Introduction
This chapter focuses on the concept of tangent to a circle and the number of tangents that can be drawn from an external point. You will learn the fundamental theorem that a tangent at any point of a circle is perpendicular to the radius through the point of contact, and that lengths of tangents drawn from an external point to a circle are equal. These properties are essential for solving problems involving circles.
Tangent to a Circle
A tangent to a circle is a line that touches the circle at exactly one point, called the point of contact. The most important property of a tangent is that it is perpendicular to the radius drawn to the point of contact. This means the angle between the tangent and the radius at the point of contact is 90 degrees. A secant is a line that intersects the circle at two points.
Key Points
- •A tangent touches the circle at exactly one point
- •Tangent is perpendicular to the radius at the point of contact
- •At any point on a circle, there is one and only one tangent
- •A line is tangent to a circle if the perpendicular distance from centre equals radius
- •No tangent can be drawn from a point inside the circle
Worked Example
A tangent PQ at point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Find PQ. Since OP is perpendicular to PQ (tangent perpendicular to radius): PQ = sqrt(OQ^2 - OP^2) = sqrt(144 - 25) = sqrt(119) cm
Watch Out
Whenever you see a tangent in a problem, immediately mark the right angle between the tangent and the radius. This is the starting point of most solutions.
Number of Tangents from a Point
The number of tangents that can be drawn to a circle from a point depends on the position of the point relative to the circle. From a point inside the circle: zero tangents. From a point on the circle: exactly one tangent. From a point outside the circle: exactly two tangents. When two tangents are drawn from an external point, they are equal in length.
Key Points
- •Point inside circle: 0 tangents
- •Point on the circle: 1 tangent
- •Point outside circle: 2 tangents
- •Tangents from an external point are equal in length
- •The line joining the external point to the centre bisects the angle between the tangents
Worked Example
Two tangents TP and TQ are drawn to a circle with centre O from an external point T. If angle PTQ = 60 degrees, find angle OPQ. TP = TQ (tangents from external point) So triangle TPQ is isosceles. angle TPQ = angle TQP = (180 - 60)/2 = 60 degrees So triangle TPQ is equilateral. angle OPT = 90 degrees (tangent perpendicular to radius) angle OPQ = 90 - 60 = 30 degrees
Properties of Tangents from an External Point
When two tangents are drawn from an external point T to a circle with centre O, touching at points P and Q, several properties hold: TP = TQ, angle OPT = angle OQT = 90 degrees, OT bisects angle PTQ, OT bisects angle POQ, and the four points O, P, T, Q form a cyclic quadrilateral (angle P + angle Q = 180 degrees).
Key Points
- •TP = TQ (equal tangent lengths)
- •OT bisects the angle between the tangents (angle PTO = angle QTO)
- •OT bisects the angle at the centre (angle POT = angle QOT)
- •angle OPT = angle OQT = 90 degrees
- •In triangle OPT: angle OTP + angle POT = 90 degrees
Worked Example
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Let O be the centre, P be the point of contact, and XY be the tangent. Assume OP is not perpendicular to XY. Then let OQ be the perpendicular from O to XY. OQ < OP (perpendicular is shortest distance). But Q lies on XY, and since OQ < OP = radius, Q is inside the circle. So XY intersects the circle at two points, contradicting that XY is a tangent. Hence OP must be perpendicular to XY.
Quick Summary
- ✓Tangent touches a circle at exactly one point
- ✓Tangent is perpendicular to the radius at the point of contact
- ✓From an external point, exactly two tangents can be drawn
- ✓Tangent lengths from an external point are equal
- ✓From a point inside a circle, no tangent can be drawn
- ✓OT bisects the angle between tangents from T
Key Formulas
Tangent perpendicular to radius: angle OPA = 90 degrees (P is point of contact)
Equal tangent lengths: If TP and TQ are tangents from T, then TP = TQ
If OQ = d, radius = r, tangent length = sqrt(d^2 - r^2)
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